Calculus on the Complex Plane C

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Calculus on the complex plane C

Let z = x + iy be the complex√ variable on the complex plane C == R × iR where i = −1. Deﬁnition A function f : C → C is holomorphic if it is complex diﬀerentiable, i.e., for each z ∈ C, f(z + h) − f(z) f0(z) = lim h→0 h exists. When f is non-constant, this is equivalent to the property: Except for isolated points, if c1, c2 are any two orthogonal curves passing through a point p ∈ C, then their image curves f ◦ c1, f ◦ c2 are orthogonal curves at f(p). This property is also equivalent to the property that the function f is angle preserving (conformal) except at isolated points, when it is not constant. Calculus on the complex plane C

Deﬁnition

A function f : M1 → M2 between two surfaces is called holomorphic if it is angle preserving except at isolated points, when 2 it is not constant. It is called meromorphic if M2 = S is the unit sphere in R3. A function f : R2 → R is called harmonic if f satisﬁes the mean value property, i.e., the value of f(p) at any point p ∈ R2, is equal to the average value of the function on every circle centered at p. Also f : R2 → R is harmonic if its Laplacian vanishes, ∆f = 0

Example If T is a temperature function in equilibrium on a domain D in the plane, then T is a harmonic function.

Theorem A harmonic function f : M → R on a surface is the real part of some holomorphic function F : M → C = R × iR. Deﬁnition of minimal surface

A surface f : M → R3 is minimal if: M has MEAN CURVATURE = 0. Small pieces have LEAST AREA. Small pieces have LEAST ENERGY. Small pieces occur as SOAP FILMS. Coordinate functions are HARMONIC. Conformal Gauss map G: M → S2 = C ∪ {∞}. MEROMORPHIC GAUSS MAP Meromorphic Gauss map Weierstrass Representation

Suppose f : M ⊂ R3 is minimal,

g: M → C ∪ {∞},

is the meromorphic Gauss map,

dh = dx3 + i ∗ dx3, is the holomorphic height diﬀerential. Then Z p 1 1 i 1 f(p) = Re − g, + g , 1 dh. 2 g 2 g Helicoid Image by Matthias Weber

M = C dh = dz = dx+i dy g(z) = eiz Catenoid Image by Matthias Weber

M = C − {(0, 0)} 1 dh = dz z g(z) = z Catenoid. Image by Matthias Weber

Key Properties:

In 1741, Euler discovered that when a catenary x1 = cosh x3 is rotated around the x3-axis, then one obtains a surface which minimizes area among surfaces of revolution after prescribing boundary values for the generating curves. In 1776, Meusnier veriﬁed that the catenoid has zero mean curvature. This surface has genus zero, two ends and total curvature −4π. Catenoid. Image by Matthias Weber

Key Properties: Together with the plane, the catenoid is the only minimal surface of revolution (Euler and Bonnet). It is the unique complete, embedded minimal surface with genus zero, finite topology and more than one end (López and Ros). The catenoid is characterized as being the unique complete, embedded minimal surface with finite topology and two ends (Schoen). Helicoid. Image by Matthias Weber

Key Properties: Proved to be minimal by Meusnier in 1776. The helicoid has genus zero, one end and inﬁnite total curvature. Together with the plane, the helicoid is the only ruled minimal surface (Catalan). It is the unique simply-connected, complete, embedded minimal surface (Meeks and Rosenberg, Colding and Minicozzi). Enneper surface. Image by Matthias Weber

Key Properties: Weierstrass Data: M = C, g(z) = z, dh = z dz. Discovered by Enneper in 1864, using his newly formulated analytic representation of minimal surfaces in terms of holomorphic data, equivalent to the Weierstrass representation. This surface is non-embedded, has genus zero, one end and total curvature −4π. It contains two horizontal orthogonal lines and the surface has two vertical planes of reﬂective symmetry. Meeks minimal M¨obius strip. Image by Matthias Weber

Key Properties:

2 z+1 Weierstrass Data: M = C − {0}, g(z) = z z−1 , z2−1 dh = i z2 dz. Found by Meeks, the minimal surface defined by this Weierstrass 3 pair double covers a complete, immersed minimal surface M1 ⊂ R which is topologically a Möbius strip. This is the unique complete, minimally immersed surface in R3 of finite total curvature −6π (Meeks). Bent helicoids. Image by Matthias Weber

Key Properties:

zn+i zn+z−n Weierstrass Data: M = C − {0}, g(z) = −z izn+i , dh = 2z dz. Discovered in 2004 by Meeks and Weber and independently by Mira. Costa torus. Image by Matthias Weber

Key Properties: Weierstrass Data: Based on the square torus 2 1 1 M = C/Z − {(0, 0), ( 2 , 0), (0, 2 )}, g(z) = P(z). Discovered in 1982 by Costa. This is a thrice punctured torus with total curvature −12π, two catenoidal ends and one planar middle end. Hoﬀman and Meeks proved its global embeddedness.

The Costa surface contains two horizontal straight lines l1, l2 that intersect orthogonally, and has vertical planes of symmetry bisecting the right angles made by l1, l2. Costa-Hoﬀman-Meeks surfaces. Image by M. Weber

Key Properties: 2 Weierstrass Data: Deﬁned in terms of cyclic covers of S .

These examples Mk generalize the Costa torus, and are complete, embedded, genus k minimal surfaces with two catenoidal ends and one planar middle end. Both existence and embeddedness were given by Hoﬀman and Meeks in 1990. Deformation of the Costa torus. Image by M. Weber

Key Properties:

The Costa surface is deﬁned on a square torus M1,1, and admits a deformation (found by Hoﬀman and Meeks, unpublished) where the planar end becomes catenoidal. Genus-one helicoid.

Key Properties: The unique end of M is asymptotic to a helicoid, so that one of the two lines contained in the surface is an axis (like in the genuine helicoid). Discovered in 1993 by Hoﬀman, Karcher and Wei. Proved embedded in 2007 by Hoﬀman, Weber and Wolf. Singly-periodic Scherk surfaces. Image by M. Weber

Key Properties: Weierstrass Data: M = (C ∪ {∞}) − {±e±iθ/2}, g(z) = z, dh = iz dz , for ﬁxed θ ∈ (0, π/2]. Q(z±e±iθ/2)

Discovered by Scherk in 1835, these surfaces denoted by Sθ form a 1-parameter family of complete, embedded, genus zero minimal surfaces in a quotient of R3 by a translation, and have four annular ends. 3 Viewed in R , each surface Sθ is invariant under reﬂection in the (x1, x3) and (x2, x3)-planes and in horizontal planes at integer heights, and can be thought of geometrically as a desingularization of two vertical planes forming an angle of θ. Doubly-periodic Scherk surfaces. Image by M. Weber

Key Properties: Weierstrass Data: M = (C ∪ {∞}) − {±e±iθ/2}, g(z) = z, dh = z dz , where θ ∈ (0, π/2] (the case θ = π . Q(z±e±iθ/2) 2 It has implicit equation ez cos y = cos x. Discovered by Scherk in 1835, are the conjugate surfaces to the singly-periodic Scherk surfaces.

. Schwarz Primitive triply-periodic surface. Image by Weber

Key Properties: Weierstrass Data: M = {(z, w) ∈ (C ∪ {∞})2 | w 2 = z 8 − 14z 4 + 1}, z dz g(z, w) = z, dh = w . Discovered by Schwarz in the 1880’s, it is also called the P-surface. This surface has a rank three symmetry group and is invariant by 3 translations in Z . Such a structure, common to any triply-periodic minimal surface (TPMS), is also known as a crystallographic cell or space tiling. Embedded TPMS divide R3 into two connected components (called labyrinths in crystallography), sharing M as boundary (or interface) and interweaving each other. Schwarz Diamond surfaces. Image by M. Weber

Discovered by Schwarz, it is the conjugate surface to the P-surface, and is another famous example of an embedded TPMS. Schoen’s triply-periodic Gyroid surface. Image by Weber In the 1960’s, Schoen made a surprising discovery: another minimal surface locally isometric to the Primitive and Diamond surface is an embedded TPMS, and named this surface the Gyroid. 1860 Riemann’s discovery! Image by Matthias Weber

Figure: Riemann minimal examples. Image by Matthias Weber

Key Properties: Discovered in 1860 by Riemann, these examples are invariant under reflection in the (x1, x3)-plane and by a translation Tλ. After appropriate scalings, they converge to catenoids as t → 0 or to helicoids as t → ∞. The Riemann minimal examples have the amazing property that every horizontal plane intersects the surface in a circle or in a line. Meeks, Pérez and Ros proved these surfaces are the only properly embedded minimal surfaces in R3 of genus 0 and infinite topology. Introduction and history of the problem

Problem: Classify all PEMS in R3 with genus zero. k = #{ends} L´opez-Ros, 1991: Finite total curvature ⇒ plane, catenoid Introduction and history of the problem

Problem: Classify all PEMS in R3 with genus zero. k = #{ends} López-Ros, 1991: Finite total curvature ⇒ plane, catenoid Collin, 1997: Finite topology and k > 1 ⇒ finite total curvature. Colding-Minicozzi, 2004: limits of simply connected minimal surfaces = minimal laminations. Meeks-Rosenberg, 2005: k = 1 ⇒ plane, helicoid. Theorem (Meeks, Pérez,Ros, 2007) k = ∞ ⇒ Riemann minimal examples. The family Rt of Riemann minimal examples Cylindrical parametrization of a Riemann minimal example 1860 Riemann’s discovery! Image by Matthias Weber

Figure: Cylindrical parametrization of a Riemann minimal example Conformal compactiﬁcation of a Riemann minimal example The moduli space of genus-zero examples Riemann minimal examples near helicoid limits A Riemann minimal example Image by Matthias Weber

Figure: 1860 Riemann’s discovery! Image by Matthias Weber

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